Locality-sensitive hashing~[Indyk,Motwani'98] is a classical data structure for approximate nearest neighbor search. It allows, after a close to linear time preprocessing of the input dataset, to find an approximately nearest neighbor of any fixed query in sublinear time in the dataset size. The resulting data structure is randomized and succeeds with high probability for every fixed query. In many modern applications of nearest neighbor search the queries are chosen adaptively. In this paper, we study the robustness of the locality-sensitive hashing to adaptive queries in Hamming space. We present a simple adversary that can, under mild assumptions on the initial point set, provably find a query to the approximate near neighbor search data structure that the data structure fails on. Crucially, our adaptive algorithm finds the hard query exponentially faster than random sampling.

翻译：局部敏感哈希[Indyk, Motwani'98]是近似最近邻搜索的经典数据结构。通过对输入数据集进行接近线性时间的预处理，该结构能够以亚线性时间复杂度为任意固定查询找到近似最近邻。所得数据结构具有随机性，且对每个固定查询能以高概率成功运行。在现代最近邻搜索的诸多应用中，查询通常是自适应选择的。本文研究了汉明空间中局部敏感哈希对自适应查询的鲁棒性。我们提出了一种简单的对抗方法，在初始点集满足温和假设的条件下，能够证明性地构造出导致近似最近邻搜索数据结构失效的查询。关键在于，我们的自适应算法发现困难查询的速度比随机采样呈指数级提升。